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  <a class="pure-menu-link nav1" onclick="animateByNav()" href="#_1">第二部分 线性代数</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#1">第1讲 行列式的性质</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1-n">1. <script type="math/tex">n</script> 行列式的几何意义</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2">2. 行列式基本公式</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3">3. 余子式和代数余子式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4">4. 拉普拉斯展开式</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#3_1">第3讲 矩阵运算</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_1">1. 矩阵</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#5">（5）分块矩阵的逆矩阵</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#6">（6）分块矩阵的伴随矩阵</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_1">2. 初等矩阵与初等变换</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_2">（1）初等矩阵</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_2">（2）初等变换（左行右列定理）</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3_2">3. 矩阵方程</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1-a">（1）若 <script type="math/tex">A</script> 可逆</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2-a_1">（2）若 <script type="math/tex">A</script> 不可逆</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#4_1">第4讲 矩阵的秩</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_3">1. 基础</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2-ab">2. 矩阵 <script type="math/tex">A,B</script> 的秩</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3_3">3. 分块矩阵的秩</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4_2">4. 转置矩阵的秩</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#5_1">5. 伴随矩阵的秩</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#6-a2">6. 与 <script type="math/tex">A^2</script> 有关的秩</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#7">7. 与基础解系有关的秩</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#8">8. 与特征值有关的秩</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#9">9. 线性表示后的秩</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1ab-a">（1）<script type="math/tex">AB</script> 的列向量可以由 <script type="math/tex">A</script> 的列向量线性表示</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2ba-a">（2）<script type="math/tex">BA</script> 的行向量可以由 <script type="math/tex">A</script> 的行向量线性表示</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#5_2">第5讲 线性方程组</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_4">1. 齐次线性方程组</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#_2">齐次线性方程组解的性质</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_3">2. 非齐次线性方程组</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_5">（1）非齐次线性方程组解的性质</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#3_4">（3）几个相关问题的等价性</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#4_3">（4）线性方程组的几何意义</a>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#6_1">第6讲 向量组</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_6">1. 向量组</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_7">（1）向量组线性相关和线性无关</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_4">（2）极大无关组</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_5">2. 向量组与向量</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3_5">3. 向量组与向量组</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1-b-a">（1）向量组 <script type="math/tex"> B </script> 能由向量组 <script type="math/tex"> A </script> 线性表示</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_6">（2）向量组等价</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4_4">4. 向量空间</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_8">（1）基</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_7">（2）坐标中的向量表示</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#3_6">（3）过渡矩阵</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#7_1">第7讲 特征值和特征向量</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_9">1. 特征值</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_8">2. 特征向量</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_10">（1）向量内积</a>
</li>
<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_9">（2）施密特正交化</a>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#8_1">第8讲 相似理论</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_11">1. 相似矩阵</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1-a-b">（1）矩阵 <script type="math/tex"> A </script> 和 <script type="math/tex">B</script> 相似的必要条件</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2-a-b">（2）矩阵 <script type="math/tex"> A </script> 和 <script type="math/tex">B</script> 相似的性质</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_10">2. 实对称矩阵</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3_7">3. 正交矩阵</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#_3">正交矩阵的性质</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4_5">4. 相似对角化</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_12">（1）充要条件</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_11">（2）充分条件</a>
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  <a class="pure-menu-link nav2" onclick="animateByNav()" href="#9_1">第9讲 二次型</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#1_13">1. 二次型</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_14">（1）标准型</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_12">（2）规范性</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#2_13">2. 线性变换和正交变换</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_15">（1）线性变换</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_14">（2）正交变换</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#3_8">3. 正负惯性指数</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4_6">4. 几何应用</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#4_7">4. 二次型标准化方法</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_16">（1）正交相似变化法</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_15">（2）拉格朗日配方法（合同）</a>
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  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#5_3">5. 正定矩阵</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#1_17">（1）前提</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#2_16">（2）充要条件</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#3_9">（3）必要条件</a>
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  <a class="pure-menu-link nav4" onclick="animateByNav()" href="#4_8">（4）重要条件</a>
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<li class="pure-menu-item">
  <a class="pure-menu-link nav3" onclick="animateByNav()" href="#6_2">6. 负定矩阵</a>
</li>

  </ul>
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<div id="content-articles">
  <h1 id="数学-线性代数" class="content-subhead">数学-线性代数</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
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    <h1 id="_1">第二部分 线性代数</h1>
<p>3Blue1Brown 线性代数的本质 ：https://www.bilibili.com/video/BV1ys411472E</p>
<h2 id="1">第1讲 行列式的性质</h2>
<h3 id="1-n">1. <script type="math/tex">n</script> 行列式的几何意义</h3>
<p>
<script type="math/tex; mode=display">
以n个向量为邻边的n维图形的体积.
</script>
</p>
<h3 id="2">2. 行列式基本公式</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
      |A^T| &= |A| \\[1ex]
|\lambda A| &= \lambda^nA \\[1ex]
       |AB| &= |A||B| = |BA|
\end{split}\end{equation}
</script>
</p>
<h3 id="3">3. 余子式和代数余子式</h3>
<p>
<script type="math/tex; mode=display">
余子式：M_{ij} \\
代数余子式：A_{ij} = (-1)^{i+j}M_{ij}
</script>
</p>
<h3 id="4">4. 拉普拉斯展开式</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\left|\begin{array}{c c}
\pmb{A} & \pmb{O} \\
\pmb{O} & \pmb{B}
\end{array}\right|
&=
\left|\begin{array}{c c}
\pmb{A} & \pmb{C} \\
\pmb{O} & \pmb{B}
\end{array}\right|
=
\left|\begin{array}{c c}
\pmb{A} & \pmb{O} \\
\pmb{C} & \pmb{B}
\end{array}\right|
=|A||B| \\[1em]
\left|\begin{array}{c c}
\pmb{O} & \pmb{A} \\
\pmb{B} & \pmb{O}
\end{array}\right|
&=
\left|\begin{array}{c c}
\pmb{C} & \pmb{A} \\
\pmb{B} & \pmb{O}
\end{array}\right|
=
\left|\begin{array}{c c}
\pmb{O} & \pmb{A} \\
\pmb{B} & \pmb{C}
\end{array}\right|
=(-1)^{mn}|A_{m \times m}||B_{n \times n}|
\end{split}\end{equation}
</script>
</p>
<h2 id="3_1">第3讲 矩阵运算</h2>
<h3 id="1_1">1. 矩阵</h3>
<h4 id="1-at">（1）转置矩阵 <script type="math/tex">A^T</script>
</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
      (A^T)^T &= A \\[1ex]
      (A+B)^T &= A^T+B^T \\[1ex]
(\lambda A)^T &= \lambda A^T \\[1ex]
       (AB)^T &= B^TA^T
\end{split}\end{equation}
</script>
</p>
<h4 id="2-a">（2）伴随矩阵 <script type="math/tex">A^*</script>
</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
 基本：AA^* &= A^*A = |A|E \\[1ex]
       A^* &= |A|A^{-1} ⟺ A^{-1} = \frac{1}{|A|}A^* \\[2em]
    (kA)^* &= k^{n-1}A^* \\[1ex]
   (A^T)^* &= (A^*)^T \\[1ex]
(A^{-1})^* &= (A^*)^{-1} = \frac{1}{|A|}A\\[1ex]
   (A^*)^* &= |A|^{n-2}A \\[2em]
重要：|A^*| &= |A|^{n-1} \\[1ex]
 |(A^*)^*| &= |A|^{(n-1)^2}
\end{split}\end{equation}
</script>
</p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-线性代数.assets/IMG_1684639F20EA-1.jpeg" alt="IMG_1684639F20EA-1" style="zoom:25%;" /></p>
<h4 id="3-a-1">（3）逆矩阵 <script type="math/tex">A^{-1}</script>
</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
  奇异矩阵 &⟺ 可逆矩阵 \\[1ex]
非奇异矩阵 &⟺ 不可逆矩阵 \\[1em]
     基本：A^{-1} &= \cfrac{1}{|A|}A^* \\[1ex]
    基本：AA^{-1} &= A^{-1}A = E \\[1ex]
(\lambda A)^{-1} &= \frac{1}{\lambda}A^{-1} \\[1ex]
       (AB)^{-1} &= B^{-1}A^{-1} \\[2em]
   (A^{-1})^{-1} &= A \\[1ex]
      (A^T)^{-1} &= (A^{-1})^T
\end{split}\end{equation}
</script>
</p>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>求逆矩阵的方法</th>
<th>公式</th>
</tr>
</thead>
<tbody>
<tr>
<td>1. 具体形式</td>
<td>
<script type="math/tex"> A^{-1}=\cfrac{E}{A} </script>
</td>
</tr>
<tr>
<td><strong>2. 初等变换</strong>（优先）</td>
<td>
<script type="math/tex"> [A｜E] = [E｜A]</script>
</td>
</tr>
<tr>
<td>3. 抽象形式</td>
<td>
<script type="math/tex"> AA^{-1}=A^{-1}A=E </script>
</td>
</tr>
</tbody>
</table></div>
<h4 id="4-an">（4）矩阵的高次方 <script type="math/tex">A^n</script>
</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
  A &= P\Lambda P^{-1} \\[1ex]
A^n &= P\Lambda P^{-1}...P\Lambda P^{-1} = P\Lambda^nP^{-1} \\[2ex]
\Lambda &= \left[\begin{array}{} 
\lambda_1 \\
&\lambda_2 \\
&&\ddots \\
&&&\lambda_n
\end{array}\right]
\end{split}\end{equation}
</script>
</p>
<h4 id="5">（5）分块矩阵的逆矩阵</h4>
<p>
<script type="math/tex; mode=display">
\left[\begin{array}{c c}
\pmb{O} & \pmb{A} \\
\pmb{B} & \pmb{O}
\end{array}\right]^{-1}
=
\left[\begin{array}{c c}
\pmb{O}^{\ \ \ \ } & \pmb{B}^{-1} \\
\pmb{A}^{-1} & \pmb{O}^{\ \ \ \ }
\end{array}\right]
</script>
</p>
<h4 id="6">（6）分块矩阵的伴随矩阵</h4>
<p>
<script type="math/tex; mode=display">
\left[\begin{array}{c c}
\pmb{O} & \pmb{A} \\
\pmb{B} & \pmb{O}
\end{array}\right]^*
=
\left|\begin{array}{c c}
\pmb{O} & \pmb{A} \\
\pmb{B} & \pmb{O}
\end{array}\right|
\left[\begin{array}{c c}
\pmb{O} & \pmb{A} \\
\pmb{B} & \pmb{O}
\end{array}\right]^{-1}
=
(-1)^{mn}|A_{m \times m}||B_{n \times n}|
\left[\begin{array}{c c}
\pmb{O}^{\ \ \ \ } & \pmb{B}^{-1} \\
\pmb{A}^{-1} & \pmb{O}^{\ \ \ \ }
\end{array}\right]
</script>
</p>
<h3 id="2_1">2. 初等矩阵与初等变换</h3>
<h4 id="1_2">（1）初等矩阵</h4>
<p>
<script type="math/tex; mode=display">
初等矩阵：由单位矩阵经过一次初等变换(行/列)得到的矩阵
</script>
</p>
<h4 id="2_2">（2）初等变换（左行右列定理）</h4>
<p>
<script type="math/tex; mode=display">
初等行变换：初等矩阵P左乘矩阵A，得PA \\[1ex]
初等列变换：初等矩阵Q右乘矩阵A，得AQ
</script>
</p>
<h3 id="3_2">3. 矩阵方程</h3>
<p>
<script type="math/tex; mode=display">
AX=B
</script>
</p>
<h4 id="1-a">（1）若 <script type="math/tex">A</script> 可逆</h4>
<p>
<script type="math/tex; mode=display">
X=A^{-1}B
</script>
</p>
<h4 id="2-a_1">（2）若 <script type="math/tex">A</script> 不可逆</h4>
<p>
<script type="math/tex; mode=display">
AX = B \\[1ex]
A[\xi_1,\xi_2,...,\xi_n] = [b_1,b_2,...,b_n] \\[1ex]
求解上述线性方程组
</script>
</p>
<h2 id="4_1">第4讲 矩阵的秩</h2>
<h3 id="1_3">1. 基础</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
0 &\le r(A_{m*n}) \le \min\{m,n\} \\[2ex]
矩阵P,Q可逆：&r(PAQ) = r(PA) = r(AQ) = r(A) \\[2em]
\end{split}\end{equation}
</script>
</p>
<h3 id="2-ab">2. 矩阵 <script type="math/tex">A,B</script> 的秩</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
r(A) + r(B) -n &\le r(A_{m\times n}B_{n\times s}) \le \min\{r(A),r(B)\} \\[2ex]
\Rightarrow 若A_{m\times n}B_{n\times s} &= O，则r(A) + r(B) \le n \\[2em]
\end{split}\end{equation}
</script>
</p>
<h3 id="3_3">3. 分块矩阵的秩</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\max\{r(\pmb{A}),r(\pmb{B})\} \le 
r\left(\left[\begin{array}{c c}
\pmb{A} & \pmb{B}\\ 
\end{array}\right]\right) 
&\le r(\pmb{A}) + r(\pmb{B}) \\[2ex]
\max\{r(\pmb{A}),r(\pmb{B})\} \le 
r\left(\left[\begin{array}{c c}
\pmb{A} \\ \pmb{B}\\ 
\end{array}\right]\right) 
&\le r(\pmb{A}) + r(\pmb{B}) \\[2em]
r(\pmb{A}) + r(\pmb{B}) \le
r\left(\left[\begin{array}{c c}
\pmb{A} & \pmb{O}\\ 
\pmb{C} & \pmb{B}
\end{array}\right]\right) 
&\le r(\pmb{A}) + r(\pmb{B}) + r(\pmb{C}) \\[2ex]
r\left(\left[\begin{array}{c c}
\pmb{A} & \pmb{O}\\ 
\pmb{O} & \pmb{B}
\end{array}\right]\right) 
&= r(\pmb{A}) + r(\pmb{B}) \\[2em]
r(\left[\pmb{A},\pmb{A}\pmb{B}\right]) &= r(\left[\pmb{A},\pmb{0}\right]) = r(\pmb{A}) \\[2ex]
r\left(\left[\begin{array}{c c c}\pmb{A} \\ \pmb{B}\pmb{A}\end{array}\right]\right) 
&= r\left(\left[\begin{array}{c c c}\pmb{A} \\ \pmb{0}\end{array}\right]\right) = r(\pmb{A})
\end{split}\end{equation}
</script>
</p>
<h3 id="4_2">4. 转置矩阵的秩</h3>
<p>
<script type="math/tex; mode=display">
r(A) = r(A^T) = r(A^T A) = r(AA^T)
</script>
</p>
<h3 id="5_1">5. 伴随矩阵的秩</h3>
<p>
<script type="math/tex; mode=display">
r(A^*) = 
\begin{cases}
n, &r(A) = n\\[1ex]
1, &r(A) = n-1\\[1ex]
0, &r(A) < n-1
\end{cases}
</script>
</p>
<blockquote class="content-quote">
<p>
<script type="math/tex; mode=display">
(1)\ 求证当\ r(A)=n\ 时\ r(A^*)=n \\[1ex]
r(AA^*)=r(A^*A)=r(|A|A^{-1}A)=r(|A|E)=n \\[1ex]
r(A)+r(A^*)-n \le r(AA^*) \le\lim\{r(A),r(A^*)\} \\[1ex]
r(A^*) \le r(AA^*) \le r(A^*) \\[1ex]
r(A^*)=r(AA^*)=r(|A|E)=n \\[2em]
(2)\ 求证当\ r(A)=n-1\ 时\ r(A^*)=1 \\[1ex]
有A的n-1阶子式不为0，A^*\neq0，r(A^*)\ge1\ \ ➊\\[1ex]
r(AA^*)=r(A^*A)=r(|A|A^{-1}A)=r(|A|E)=0 \\[1ex]
r(A)+r(A^*)-n \le r(AA^*) \le\lim\{r(A),r(A^*)\} \\[1ex]
r(A^*)-1 \le r(AA^*)=0 \\[1ex]
r(A^*)\le1\ \ ➋\\[1ex]
由\ ➊➋\ 得r(A^*)=1 \\[2em]
(3)\ 求证当\ r(A)\lt n-1\ 时\ r(A^*)=0 \\[1ex]
r(A)+r(A^*)-n \le r(AA^*) \le\lim\{r(A),r(A^*)\} \\[1ex]
r(A^*)-1 \lt r(AA^*)=0 \\[1ex]
r(A^*)=0
</script>
</p>
</blockquote>
<h3 id="6-a2">6. 与 <script type="math/tex">A^2</script> 有关的秩</h3>
<p>
<script type="math/tex; mode=display">
若\ A^2=A,\ \ \ 则\ r(A\ \ \ \ \ \ \ \ )+r(A-E)=n \\[1ex]
若\ A^2=E,\ \ \ 则\ r(A+E)+r(A-E)=n
</script>
</p>
<h3 id="7">7. 与基础解系有关的秩</h3>
<p>
<script type="math/tex; mode=display">
\pmb{A}\pmb{x}=\pmb{0}\ 的基础解系所含向量的个数\ s=n-r(\pmb{A})
</script>
</p>
<h3 id="8">8. 与特征值有关的秩</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
若\ A\sim\Lambda,\ \ \ &则\ n_i = n-r(\lambda_i E -A),\ \ \ 其中\ \lambda_i\ 是\ n_i\ 重根 \\[2ex]
&r(A) = 非0特征值的个数
\end{split}\end{equation}
</script>
</p>
<h3 id="9">9. 线性表示后的秩</h3>
<h4 id="1ab-a">（1）<script type="math/tex">AB</script> 的列向量可以由 <script type="math/tex">A</script> 的列向量线性表示</h4>
<p>另外，<script type="math/tex">AB</script> 表示矩阵 <script type="math/tex">A</script> 经过矩阵 <script type="math/tex">B</script> 对应的 <strong>列变换</strong> 后的矩阵<br />
<script type="math/tex; mode=display">
r(\left[\pmb{A},\pmb{A}\pmb{B}\right]) = r(\left[\pmb{A},\pmb{0}\right]) = r(\pmb{A})
</script>
</p>
<h4 id="2ba-a">（2）<script type="math/tex">BA</script> 的行向量可以由 <script type="math/tex">A</script> 的行向量线性表示</h4>
<p>另外，<script type="math/tex">BA</script> 表示矩阵 <script type="math/tex">A</script> 经过矩阵 <script type="math/tex">B</script> 对应的 <strong>行变换</strong> 后的矩阵<br />
<script type="math/tex; mode=display">
r\left(\left[\begin{array}{c c c}\pmb{A} \\ \pmb{B}\pmb{A}\end{array}\right]\right) 
= r\left(\left[\begin{array}{c c c}\pmb{A} \\ \pmb{0}\end{array}\right]\right)   = r(\pmb{A})
</script>
</p>
<h2 id="5_2">第5讲 线性方程组</h2>
<h3 id="1_4">1. 齐次线性方程组</h3>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>
<script type="math/tex"> Ax=0 </script>
</th>
<th>公式</th>
<th>通解</th>
</tr>
</thead>
<tbody>
<tr>
<td>有唯一零解</td>
<td>
<script type="math/tex"> r(A) = n </script>
</td>
<td></td>
</tr>
<tr>
<td>有无限多解</td>
<td>
<script type="math/tex"> r(A)\lt n </script>
</td>
<td>
<script type="math/tex">x=k_1\xi_1+k_2\xi_2+\cdots+k_{n-r}\xi_{n-r}</script>
</td>
</tr>
</tbody>
</table></div>
<h4 id="_2">齐次线性方程组解的性质</h4>
<ol>
<li>若 <script type="math/tex"> \pmb{\xi}_1,\pmb{\xi}_2 </script> 为 <script type="math/tex"> \pmb{A}\pmb{x}=\pmb{0} </script> 的解，则 <script type="math/tex"> \pmb{\xi}_1+\pmb{\xi}_2 </script> 也为其解</li>
<li>若 <script type="math/tex"> \pmb{\xi} </script> 为 <script type="math/tex"> \pmb{A}\pmb{x}=\pmb{0} </script> 的解，则 <script type="math/tex"> k\pmb{\xi} </script> 也为其解</li>
<li>
<script type="math/tex"> \pmb{A}\pmb{x}=\pmb{0} </script> 的解集的秩为 <script type="math/tex"> R_s=n-r(\pmb{A})=c </script> （<script type="math/tex">c</script> 为最大无关组的个数）</li>
</ol>
<h3 id="2_3">2. 非齐次线性方程组</h3>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>
<script type="math/tex"> Ax=b </script>
</th>
<th>公式</th>
<th>通解</th>
</tr>
</thead>
<tbody>
<tr>
<td>无解</td>
<td>
<script type="math/tex"> r(A)\lt r(A,b) </script>
</td>
<td></td>
</tr>
<tr>
<td>有唯一解</td>
<td>
<script type="math/tex"> r(A) = r(A,b) = n </script>
</td>
<td>
<script type="math/tex">x=\eta^*</script>
</td>
</tr>
<tr>
<td>有无限多解</td>
<td>
<script type="math/tex"> r(A) = r(A,b) \lt n </script>
</td>
<td>
<script type="math/tex">x=\eta^*+k_1\xi_1+k_2\xi_2+\cdots+k_{n-r}\xi_{n-r}</script>
</td>
</tr>
</tbody>
</table></div>
<h4 id="1_5">（1）非齐次线性方程组解的性质</h4>
<ol>
<li>
<p>若 <script type="math/tex"> \pmb{\eta}_1,\ \pmb{\eta}_2 </script> 为 <script type="math/tex"> \pmb{A}\pmb{x}=\pmb{b} </script> 的解，则 <script type="math/tex">\pmb{\eta}_1-\pmb{\eta}_2</script> 为 <script type="math/tex">\pmb{A}\pmb{x}=\pmb{0}</script> 的解</p>
</li>
<li>
<p>
<script type="math/tex">\pmb{\xi}_1,\pmb{\xi}_2,\cdots,\pmb{\xi}_{n-r}</script> 为 <script type="math/tex">\pmb{A}\pmb{x}=\pmb{0}</script> 的基础解系，<script type="math/tex">\pmb{\eta}^*</script> 为 <script type="math/tex">\pmb{A}\pmb{x}=\pmb{b}</script> 的一个特解，则其通解为 <script type="math/tex">\pmb{\eta}^*+k_1\pmb{\xi}_1+k_2\pmb{\xi}_2+\cdots+k_{n-r}\pmb{\xi}_{n-r}</script>
</p>
</li>
<li>
<p>
<script type="math/tex">\pmb{\alpha}_1,\pmb{\alpha}_2,\cdots,\pmb{\alpha}_s</script> 为 <script type="math/tex">\pmb{A}\pmb{x}=\pmb{b}</script> 的一组特解，则 <script type="math/tex">k_1\pmb{\alpha}_1+k_2\pmb{\alpha}_2+\cdots+k_s\pmb{\alpha}_s</script> 为 <script type="math/tex">\pmb{A}\pmb{x}=\pmb{b}</script> 的解的 <strong>充要条件</strong> 是 <script type="math/tex">k_1+k_2+\cdots+k_s=1</script>
</p>
</li>
<li>
<p>
<script type="math/tex">\pmb{\alpha}_1,\pmb{\alpha}_2,\cdots,\pmb{\alpha}_s</script> 为 <script type="math/tex">\pmb{A}\pmb{x}=\pmb{b}</script> 的一组特解，则 <script type="math/tex">k_1\pmb{\alpha}_1+k_2\pmb{\alpha}_2+\cdots+k_s\pmb{\alpha}_s</script> 为 <script type="math/tex">\pmb{A}\pmb{x}=\pmb{0}</script> 的解的 <strong>充要条件</strong> 是 <script type="math/tex">k_1+k_2+\cdots+k_s=0</script>
</p>
</li>
</ol>
<p>​   </p>
<h4 id="2-axb">（2）克拉默法则 <script type="math/tex">Ax=b</script>
</h4>
<p>以 3 阶举例<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
x_1 =
\cfrac{
\left|\begin{array}{c c c}
b_1 & a_{12} & a_{13} \\
b_2 & a_{22} & a_{23} \\
b_3 & a_{32} & a_{33}
\end{array}\right|
}{
\left|\begin{array}{c c c}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right|
},
x_2 =
\cfrac{
\left|\begin{array}{c c c}
a_{11} & b_1 & a_{13} \\
a_{21} & b_2 & a_{23} \\
a_{31} & b_3 & a_{33}
\end{array}\right|
}{
\left|\begin{array}{c c c}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right|
},
x_3 =
\cfrac{
\left|\begin{array}{c c c}
a_{11} & a_{12} & b_1 \\
a_{21} & a_{22} & b_2 \\
a_{31} & a_{32} & b_3
\end{array}\right|
}{
\left|\begin{array}{c c c}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right|
}
\end{split}\end{equation}
</script>
</p>
<h4 id="3_4">（3）几个相关问题的等价性</h4>
<ol>
<li>方程组 <script type="math/tex">\pmb{A}\pmb{x}=\pmb{b}</script> 有解</li>
<li>向量 <script type="math/tex">\pmb{b}</script> 能由向量组 <script type="math/tex">\pmb{a}_1, \pmb{a}_2, \cdots,\pmb{a}_n</script> 线性表示</li>
<li>向量组 <script type="math/tex">\pmb{a}_1, \pmb{a}_2, \cdots,\pmb{a}_n</script> 与向量组 <script type="math/tex">\pmb{a}_1, \pmb{a}_2, \cdots, \pmb{a}_n, \pmb{b}</script>  等价</li>
<li>
<script type="math/tex">r(\pmb{A}) = r(\pmb{A},\pmb{b})</script>
</li>
</ol>
<h4 id="4_3">（4）线性方程组的几何意义</h4>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-线性代数.assets/img.png" alt="img" style="zoom:20%;" /></p>
<p><img class="pure-img" alt="imgtabla" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-线性代数.assets/imgtabla.png" /></p>
<h2 id="6_1">第6讲 向量组</h2>
<h3 id="1_6">1. 向量组</h3>
<h4 id="1_7">（1）向量组线性相关和线性无关</h4>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>向量组</th>
<th>线性相关</th>
<th>线性无关</th>
</tr>
</thead>
<tbody>
<tr>
<td>向量个数 <script type="math/tex">n>m</script> 维数</td>
<td></td>
<td></td>
</tr>
<tr>
<td>向量个数 <script type="math/tex">n=m</script> 维数</td>
<td>
<script type="math/tex">\lvert\pmb{\alpha_1},\pmb{\alpha_2},\cdots,\pmb{\alpha_n}\rvert=0</script>
</td>
<td>
<script type="math/tex">\lvert\pmb{\alpha_1},\pmb{\alpha_2},\cdots,\pmb{\alpha_n}\rvert\neq0</script>
</td>
</tr>
<tr>
<td>向量个数 <script type="math/tex">n<m</script> 维数</td>
<td>
<script type="math/tex"> r(A) \lt m </script>
</td>
<td>
<script type="math/tex"> r(A) = m </script>
</td>
</tr>
</tbody>
</table></div>
<h4 id="2_4">（2）极大无关组</h4>
<p>定义：若 <script type="math/tex"> r(A) = r </script> ，则 <script type="math/tex"> r+1 </script> 个向量都线性无关，这 <script type="math/tex"> r </script> 个向量组成的向量组 <script type="math/tex"> A_0 </script> 称为 <script type="math/tex"> A </script> 的一个极大无关组。</p>
<p>求极大无关组</p>
<ul>
<li>将矩阵 <script type="math/tex">A</script> 初等行变换成阶梯形矩阵 <script type="math/tex">B</script>
</li>
<li>在 <script type="math/tex">B</script> 中找出一个秩为 <script type="math/tex">r(B)</script> 的子向量组，其对应的 <script type="math/tex">A</script> 中的子向量组就是极大无关组。</li>
</ul>
<h3 id="2_5">2. 向量组与向量</h3>
<p>非齐次方程组<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(\pmb{\alpha_1},\pmb{\alpha_2},\cdots,\pmb{\alpha_n})\pmb{x}=\pmb{\beta} \\[2ex]
\pmb{A}\pmb{x}=\pmb{\beta}
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&r(\pmb{A})<r(\pmb{A},\pmb{\beta}) \ \ \ \ \ \ \ \  ⟺ 无解 \ \ \ \ \ \ \ \ \ \ ⟺ 不能表示\\[2ex]
&r(\pmb{A})=r(\pmb{A},\pmb{\beta})=n ⟺ 有唯一解 \ \ \ ⟺ 有唯一表示方法\\[2ex]
&r(\pmb{A})=r(\pmb{A},\pmb{\beta})<n ⟺ 有无穷多解 ⟺ 有无穷多表示法
\end{split}\end{equation}
</script>
</p>
<h3 id="3_5">3. 向量组与向量组</h3>
<h4 id="1-b-a">（1）向量组 <script type="math/tex"> B </script> 能由向量组 <script type="math/tex"> A </script> 线性表示</h4>
<ul>
<li>向量组 <script type="math/tex"> B </script> 中的每一个向量 <script type="math/tex"> b_i </script> 都能由向量组 <script type="math/tex"> A=(a_1,a_2,...,a_n) </script> 线性表示， <script type="math/tex"> r(A) = r(A,B)\ 或\ r(A) > r(B) </script>
</li>
</ul>
<h4 id="2_6">（2）向量组等价</h4>
<ul>
<li>定义：向量组 <script type="math/tex"> B </script> 能由向量组 <script type="math/tex"> A </script> 线性表示，向量组 <script type="math/tex"> A </script> 也能由向量组 <script type="math/tex"> B </script> 线性表示</li>
<li>记号：<script type="math/tex"> A ≅ B</script>
</li>
<li>前提要求：向量组 <script type="math/tex">A,B</script> 同维度，不要求向量个数相等</li>
<li>充要条件： <script type="math/tex"> r(A) = r(B) = r(A,B) </script>
</li>
</ul>
<h3 id="4_4">4. 向量空间</h3>
<h4 id="1_8">（1）基</h4>
<p>向量组 <script type="math/tex"> \alpha_1,\alpha_2,\alpha_3 </script> 为 <script type="math/tex"> R^3 </script> 的一个<strong>基</strong>，则 <script type="math/tex"> \alpha_1,\alpha_2,\alpha_3 </script>
<strong>线性无关</strong>：<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
r(\alpha_1,\alpha_2,\alpha_3) &= 3 \\[2ex]
|(\alpha_1,\alpha_2,\alpha_3)| &= 0
\end{split}\end{equation}
</script>
</p>
<h4 id="2_7">（2）坐标中的向量表示</h4>
<p><strong>非零向量</strong> <script type="math/tex"> \xi </script> 在基 <script type="math/tex"> \alpha_1,\alpha_2,\alpha_3 </script> 下的表达式：<br />
<script type="math/tex; mode=display">
\xi = k_1\alpha_1 + k_2\alpha_2 + k_3\alpha_3
</script>
</p>
<h4 id="3_6">（3）过渡矩阵</h4>
<p>
<script type="math/tex; mode=display">
[\alpha_1,\alpha_2,\alpha_3]C=[\beta_1,\beta_2,\beta_3] \\[2em]
C\ 称为基\ [\alpha_1,\alpha_2,\alpha_3]\ 到\ [\beta_1,\beta_2,\beta_3]\ 的过渡矩阵 \\[2ex]
实际是向量空间\ [\alpha_1,\alpha_2,\alpha_3]\ 通过矩阵\ C\ 变换映射到另一个向量空间\ [\beta_1,\beta_2,\beta_3]
</script>
</p>
<h2 id="7_1">第7讲 特征值和特征向量</h2>
<h3 id="1_9">1. 特征值</h3>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
A\xi&=\lambda \xi,\ \xi\neq0 \\[3ex]
|\lambda_0E-A|&=0 \\[1ex]
|A|&=\lambda_1\lambda_2...\lambda_n \\[1ex]
tr(A)&=\lambda_1+\lambda_2...+\lambda_n \\[3ex]
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
A\ 的特征值与\ A^T\ 相同，但特征向量不同
</script>
</p>
<ul>
<li>判断 <script type="math/tex">\lambda</script> 是不是 <script type="math/tex">A</script> 的特征值，若 <script type="math/tex">|\lambda E-A|=0</script>  则是，反之则不是.</li>
</ul>
<h3 id="2_8">2. 特征向量</h3>
<p>几何意义：向量空间变换后方向不变的向量。</p>
<blockquote class="content-quote">
<p>3Blue1Brown 特征向量与特征值 https://www.bilibili.com/video/BV1ys411472E?p=14</p>
</blockquote>
<p>
<script type="math/tex; mode=display">
带入\ \lambda_0，得\ [\lambda_0\pmb{E}-\pmb{A}]x=0，求特征向量 \\[2ex]
\pmb{\xi}\ 是属于\ \lambda_0\ 的特征向量，\pmb{\xi}\ 是\ [\lambda_0\pmb{E}-\pmb{A}]x=0\ 的非零解\ (\pmb{\xi}\neq0)
</script>
</p>
<p>重要结论：</p>
<ul>
<li>
<script type="math/tex">k</script> 重特征值，至多有 <script type="math/tex">k</script> 个线性无关的特征向量</li>
<li>
<script type="math/tex">\xi_1,\xi_2</script> 是 <script type="math/tex">A</script> 的属于不同特征值 <script type="math/tex">\lambda_1,\lambda_2</script> 的特征向量，则 <script type="math/tex">\xi_1,\xi_2</script> 线性无关</li>
<li>
<script type="math/tex">\xi_1,\xi_2</script> 是 <script type="math/tex">A</script> 的属于不同特征值 <script type="math/tex">\lambda_1,\lambda_2</script> 的特征向量，则 <script type="math/tex">k_1\xi_1+k_2\xi_2(k_1,k_2\neq0)</script> 不是 <script type="math/tex">A</script> 的特征向量</li>
<li>
<script type="math/tex">\xi_1,\xi_2</script> 是 <script type="math/tex">A</script> 的特征值 <script type="math/tex">\lambda</script> 的特征向量，则 <script type="math/tex">k_1\xi_1+k_2\xi_2(k_1,k_2不同时为0)</script> 仍是 <script type="math/tex">\lambda</script> 的特征向量</li>
</ul>
<h4 id="1_10">（1）向量内积</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
[x,y] &= x^Ty \\[1ex]
||x|| &= \sqrt{[x,x]} = \sqrt{x_1^2+x_2^2+...+x_n^2} \\[1ex]
0\le\theta&=\arccos\cfrac{[x,y]}{||x||*||y||}\le\pi \\[1ex]
[x,y] &= ||x||*||y||*\cos\theta
\end{split}\end{equation}
</script>
</p>
<ul>
<li>向量正交： <script type="math/tex"> \{[x,y] = 0\} \Rightarrow \{向量x和y正交\} </script>
</li>
</ul>
<h4 id="2_9">（2）施密特正交化</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\beta_1&=\alpha_1 \\
\beta_2&=\alpha_2-\cfrac{[\beta_1,\alpha_2]}{[\beta_1,\beta_1]}\beta_1 \\
\ ...\ &...\ ... \\
\beta_r&=\alpha_r-\cfrac{[\beta_1,\alpha_r]}{[\beta_1,\beta_1]}\beta_1-\cfrac{[\beta_2,\alpha_r]}{[\beta_2,\beta_2]}\beta_2-...-\cfrac{[\beta_{r-1},\alpha_r]}{[\beta_{r-1},\beta_{r-1}]}\beta_{r-1}
\end{split}\end{equation}
</script>
</p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-线性代数.assets/截屏2020-12-07 00.34.40.jpg" alt="截屏2020-12-07 00.34.40" style="zoom:50%;" /></p>
<blockquote class="content-quote">
<p>【2021.6】<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
已知\ \alpha_1&=(1,0,1)^T \\[1ex]
\alpha_2&=(1,2,1)^T \\[1ex]
\alpha_3&=(3,1,2)^T \\[2ex]
记\ \beta_1&=\alpha_1 \\[1ex]
\beta_1&=\alpha_1-k\beta_1 \\[1ex]
\beta_1&=\alpha_1-l_1\beta_1-l_2\beta_2 \\[2ex]
且\ \beta_1,\beta_2,\beta_3\ &两两正交，求\ k,l_1,l_2
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
根据施密特正交法\ \ k&=\cfrac{[\beta_1,\alpha_2]}{[\beta_1,\beta_1]} \\[1ex]
l_1&=\cfrac{[\beta_1,\alpha_3]}{[\beta_1,\beta_1]},
l_2=\cfrac{[\beta_2,\alpha_3]}{[\beta_2,\beta_2]}
\end{split}\end{equation}
</script>
</p>
</blockquote>
<h2 id="8_1">第8讲 相似理论</h2>
<h3 id="1_11">1. 相似矩阵</h3>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>关系</th>
<th>表达式</th>
<th>矩阵要求</th>
<th>记号</th>
<th>秩</th>
<th>正负惯性指数</th>
<th>特征值</th>
<th>关系</th>
</tr>
</thead>
<tbody>
<tr>
<td>等价</td>
<td>
<script type="math/tex"> P^{-1}AQ=B_1 </script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，可逆矩阵 <script type="math/tex"> P </script>，可逆矩阵 <script type="math/tex">Q</script>
</td>
<td>
<script type="math/tex"> A ≅ B_1 </script>
</td>
<td>相同</td>
<td></td>
<td></td>
<td>最弱</td>
</tr>
<tr>
<td>合同</td>
<td>
<script type="math/tex"> P^TAP=B_2 </script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，可逆矩阵 <script type="math/tex"> P </script>
</td>
<td>
<script type="math/tex"> A≃ B_2 </script>
</td>
<td>相同</td>
<td>相同</td>
<td></td>
<td></td>
</tr>
<tr>
<td>相似</td>
<td>
<script type="math/tex"> P^{-1}AP=B_3 </script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，可逆矩阵 <script type="math/tex"> P </script>
</td>
<td>
<script type="math/tex"> A \sim B_3 </script>
</td>
<td>相同</td>
<td>相同</td>
<td>相同</td>
<td>最强</td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>对角化</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>相似</td>
<td>
<script type="math/tex">P^{-1}AP=\Lambda</script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，可逆矩阵 <script type="math/tex"> P </script>
</td>
<td>
<script type="math/tex"> A\sim \Lambda </script>
</td>
<td>相同</td>
<td>相同</td>
<td>相同</td>
<td></td>
</tr>
<tr>
<td>相似</td>
<td>
<script type="math/tex">Q^{-1}AQ=\Lambda</script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，<strong>正交</strong>矩阵 <script type="math/tex"> Q </script>
</td>
<td>
<script type="math/tex"> A\sim \Lambda </script>
</td>
<td>相同</td>
<td>相同</td>
<td>相同</td>
<td></td>
</tr>
<tr>
<td>合同相似</td>
<td>
<script type="math/tex">Q^{T}AQ=\Lambda</script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，<strong>正交</strong>矩阵 <script type="math/tex"> Q </script>
</td>
<td>
<script type="math/tex"> A\sim \Lambda </script>
</td>
<td>相同</td>
<td>相同</td>
<td>相同</td>
<td></td>
</tr>
<tr>
<td>合同</td>
<td>
<script type="math/tex">C^{T}AC=\Lambda</script>
</td>
<td>任意实对称矩阵 <script type="math/tex">A</script>，可逆矩阵 <script type="math/tex"> C </script>
</td>
<td>
<script type="math/tex"> A≃ \Lambda </script>
</td>
<td>相同</td>
<td>相同</td>
<td></td>
<td></td>
</tr>
</tbody>
</table></div>
<blockquote class="content-quote">
<p>正(负)惯性指数：线性代数里矩阵的 <strong>正(负)的特征值个数</strong>。</p>
</blockquote>
<h4 id="1-a-b">（1）矩阵 <script type="math/tex"> A </script> 和 <script type="math/tex">B</script> 相似的必要条件</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
|A|&=|B| \\[1ex]
r(A)&=r(B) \\[2ex]
tr(A)&=tr(B) \\[1ex]
\lambda_A&=\lambda_B \\[1ex]
|\lambda E - A|&=|\lambda E - B| \\[1ex]
r(\lambda E - A)&=r(\lambda E - B)
\end{split}\end{equation}
</script>
</p>
<p>
<script type="math/tex">(\lambda_iE-A)x=0\ </script>和<script type="math/tex">\ (\lambda_iE-B)x=0\ </script>的解中线性无关向量的个数相等</p>
<p>
<script type="math/tex; mode=display">
n-r(\lambda_iE-A)=n-r(\lambda_iE-B)
</script>
</p>
<h4 id="2-a-b">（2）矩阵 <script type="math/tex"> A </script> 和 <script type="math/tex">B</script> 相似的性质</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(1)\ A\sim B &\Rightarrow A^T\sim B^T \\[1ex] 
             &\Rightarrow A^{-1}\sim B^{-1} \\[1ex] 
             &\Rightarrow A^*\sim B^* \\[1ex]
             &\Rightarrow A^m\sim B^m \\[1ex] 
             &\Rightarrow f(A)\sim f(B) \\[2em]
(2)\ A\sim B,\ B\sim C &\Rightarrow A\sim C \\[2em]
(3)\ A\sim B &\Rightarrow A+A^T \nsim B + B^T \\[1ex]
       &\Rightarrow A+A^{-1} \sim B + B^{-1}\\[1ex]
       &(2016考研数学一选择题5) \\[2em]
(4)\ A\sim B,\ C\sim D &\Rightarrow 
\left[\begin{array}{c c}
A & O \\
O & B
\end{array}\right]
\sim
\left[\begin{array}{c c}
C & O \\
O & D
\end{array}\right]
\end{split}\end{equation}
</script>
</p>
<h3 id="2_10">2. 实对称矩阵</h3>
<p>定义：矩阵 <script type="math/tex">A</script> 的元素都为实数，且 <script type="math/tex">A^T = A</script>
</p>
<p>主要性质：</p>
<ul>
<li>实对称矩阵 <script type="math/tex">A</script> 的特征值均为实数，特征向量均为实向量</li>
<li>实对称矩阵 <script type="math/tex">A</script> 的不同特征值对应的特征向量都是正交的</li>
<li>实对称矩阵 <script type="math/tex">A</script> 一定可以用 <strong>正交矩阵</strong> 相似对角化（<script type="math/tex">Q^TAQ=Q^{-1}AQ=\Lambda</script>），且 <script type="math/tex">\Lambda</script> 对角线上的元素为矩阵 <script type="math/tex">A</script> 的特征值</li>
<li>若实对称矩阵 <script type="math/tex">A</script> 有有 <script type="math/tex">k</script> 重特征值 <script type="math/tex">\lambda_0</script>，则必定有 <script type="math/tex">k</script> 个线性无关的特征向量，且  <script type="math/tex">r(\lambda_0E-A)=n-k</script>
</li>
</ul>
<h3 id="3_7">3. 正交矩阵</h3>
<p>
<script type="math/tex; mode=display">
Q=(\alpha_1,\alpha_2,\cdots,\alpha_n)\ 为正交矩阵 \\[2ex]
\alpha_1,\alpha_2,\cdots,\alpha_n\ 两两正交，且均为单位矩阵
</script>
</p>
<h4 id="_3">正交矩阵的性质</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
Q\ 为正交矩阵&⟺Q^TQ=E \\
&⟺QQ^T=E \\[2ex]
(重要)&⟺Q^{-1}=Q^T \\[2ex]
&⟺Q&由规范正交基组成 \\
&⟺Q^T&为正交矩阵 \\
&⟺Q^{-1}&为正交矩阵 \\
&⟺Q^*&为正交矩阵 \\
&⟺-Q&为正交矩阵 \\[2ex]
Q_1,Q_2\ 为同阶正交矩阵&⟺Q_1Q_2&为正交矩阵
\end{split}\end{equation}
</script>
</p>
<h3 id="4_5">4. 相似对角化</h3>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>关系</th>
<th>表达式</th>
<th>矩阵要求</th>
<th>记号</th>
</tr>
</thead>
<tbody>
<tr>
<td>相似</td>
<td>
<script type="math/tex">P^{-1}AP=\Lambda</script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，可逆矩阵 <script type="math/tex"> P </script>
</td>
<td>
<script type="math/tex"> A \sim \Lambda </script>
</td>
</tr>
<tr>
<td>相似</td>
<td>
<script type="math/tex">Q^{-1}AQ=\Lambda</script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，<strong>正交</strong>矩阵 <script type="math/tex"> Q </script>
</td>
<td>
<script type="math/tex"> A \sim \Lambda </script>
</td>
</tr>
<tr>
<td>合同相似</td>
<td>
<script type="math/tex">Q^{T}AQ=\Lambda</script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，<strong>正交</strong>矩阵 <script type="math/tex"> Q </script>
</td>
<td>
<script type="math/tex"> A \sim \Lambda </script>
</td>
</tr>
</tbody>
</table></div>
<p><strong>定义：</strong>设 <script type="math/tex"> n </script> 阶矩阵 <script type="math/tex"> A </script> ，若存在 <script type="math/tex"> n </script> 阶可逆矩阵 <script type="math/tex"> P </script> 和正交矩阵 <script type="math/tex">Q</script> 使得， <script type="math/tex"> P^{-1}AP=Q^TAQ=\Lambda </script> ，其中 <script type="math/tex"> \Lambda </script> 是对角矩阵，则称可相似对角化，记 <script type="math/tex"> A\sim\Lambda </script> ，则称 <script type="math/tex"> \Lambda </script> 是 <script type="math/tex"> A </script> 的相似对角化.</p>
<p>（1）矩阵 <script type="math/tex">A</script> 的 <u>正交</u> 特征向量矩阵 <script type="math/tex">P</script> (<strong>是可逆矩阵，不一定是正交矩阵</strong>) 可以将矩阵 <script type="math/tex">A</script> 相似对角化：<br />
<script type="math/tex; mode=display">
\ P = (\xi_1,\xi_2,...,\xi_n) \\[2ex]
\xi_n为A的\underline{正交}特征向量(线性无关)，若特征向量不正交，需要施密特正交化
</script>
</p>
<p>
<script type="math/tex; mode=display">
(相似)\ \ \ P^{-1}AP=\Lambda=
\left[\begin{array}{c c c}
\lambda_1&& \\
&\ddots& \\
&&\lambda_n 
\end{array}\right]
</script>
</p>
<p>（2）矩阵 <script type="math/tex">A</script> 的 <u>单位化正交</u> 特征向量矩阵 <script type="math/tex">Q</script> (<strong>一定是正交矩阵</strong>) 可以将矩阵 <script type="math/tex">A</script> 合同相似对角化：<br />
<script type="math/tex; mode=display">
正交矩阵\ Q = (\eta_1,\eta_2,...,\eta_n) \\[2ex] 
\eta_n为\alpha_n的\underline{单位化正交}特征向量(线性无关)，若特征向量不正交，需要施密特正交化，再单位化
</script>
</p>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(合同相似)\ \ \ Q^{T}AQ &= \Lambda =
\left[\begin{array}{c c c}
\lambda_1&& \\
&\ddots& \\
&&\lambda_n 
\end{array}\right]\\
(相似)\ \ \ Q^{-1}AQ &= \Lambda \\[2ex]
(特征向量&单位化之后依然为特征向量)
\end{split}\end{equation}
</script>
</p>
<h4 id="1_12">（1）充要条件</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
A\ 有\ n\ 个线性无关的特征向量 &⟺\ A\sim\Lambda \\[1ex]
n_i=n-r(\lambda_iE-A) &⟺\ A\sim\Lambda
\end{split}\end{equation}
</script>
</p>
<h4 id="2_11">（2）充分条件</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
A \ 是实对称矩阵 \ ⟹ A \sim \Lambda \\[1ex]
A \ 有 \ n \ 个互异特征值 \ ⟹ A \sim \Lambda \\[1ex]
A^2 = A ⟹ A \sim \Lambda \\[1ex]
A^2 = E ⟹ A \sim \Lambda \\[1ex]
\end{split}\end{equation}
</script>
</p>
<h2 id="9_1">第9讲 二次型</h2>
<h3 id="1_13">1. 二次型</h3>
<p>
<script type="math/tex; mode=display">
f(\pmb{x})=\pmb{x}^T\pmb{A}\pmb{x}
</script>
</p>
<h4 id="1_14">（1）标准型</h4>
<p>
<script type="math/tex; mode=display">
f(\pmb{x})=d_1 x_1^2+d_2 x_2^2+\cdots+d_n x_n^2 \\[2ex]
二次型只有平方项，没有交叉项
</script>
</p>
<h4 id="2_12">（2）规范性</h4>
<p>
<script type="math/tex; mode=display">
f(\pmb{x})=d_1 x_1^2+d_2 x_2^2+\cdots+d_n x_n^2,\ \ (d_1,d_2,\cdots,d_n=-1,0,1)\\[2ex]
二次型只有平方项，且平方项的系数为\ -1\ 或\ 0\ 或\ 1
</script>
</p>
<h3 id="2_13">2. 线性变换和正交变换</h3>
<h4 id="1_15">（1）线性变换</h4>
<p>
<script type="math/tex; mode=display">
\pmb{x}=\pmb{C}\pmb{y} \\[1ex]
若\ \pmb{C}\ 可逆，则称可逆线性变换
</script>
</p>
<p>任意二次型（实对称矩阵 <script type="math/tex">A</script>），总有线性变换  <script type="math/tex">\pmb{x}=\pmb{C}\pmb{y}</script>  使之称为 <strong>标准型</strong> 或 <strong>规范型</strong>（合同对角化）：<br />
<script type="math/tex; mode=display">
\pmb{C}^{T}\pmb{A}\pmb{C}=\pmb{\Lambda} \\[1ex]
\pmb{x}^T\pmb{A}\pmb{x}=\pmb{y}^T(\pmb{C}^T\pmb{A}\pmb{C})\pmb{y}=\pmb{y}^T\pmb{\Lambda}\pmb{y}
</script>
</p>
<h4 id="2_14">（2）正交变换</h4>
<p>
<script type="math/tex; mode=display">
\pmb{x}=\pmb{Q}\pmb{y} \\[1ex]
\pmb{Q}\ 为正交矩阵
</script>
</p>
<ul>
<li>任意二次型（实对称矩阵 <script type="math/tex">A</script>），总有正交变换  <script type="math/tex">\pmb{x}=\pmb{Q}\pmb{y}</script>  使之称为 <strong>标准型</strong>（合同相似对角化）：</li>
</ul>
<p>
<script type="math/tex; mode=display">
\pmb{C}^{T}\pmb{A}\pmb{C}=\pmb{\Lambda} \\[1ex]
\pmb{x}^T\pmb{A}\pmb{x}=\pmb{y}^T(\pmb{Q}^T\pmb{A}\pmb{Q})\pmb{y}=\pmb{y}^T\pmb{\Lambda}\pmb{y} \\[2em]
\pmb{C}^{-1}\pmb{A}\pmb{C}=\pmb{\Lambda} \\[1ex]
\pmb{x}^T\pmb{A}\pmb{x}=\pmb{y}^T(\pmb{Q}^{-1}\pmb{A}\pmb{Q})\pmb{y}=\pmb{y}^T\pmb{\Lambda}\pmb{y}
</script>
</p>
<blockquote class="content-quote">
<p>
<script type="math/tex; mode=display">
正交变换的性质(内积不变)：||\pmb{x}||=\sqrt{\pmb{x}^T\pmb{x}}=\sqrt{\pmb{y}^T\pmb{Q}^T\pmb{Q}\pmb{y}}=||\pmb{y}||
</script>
</p>
<p>在线性代数中，正交变换是线性变换的一种，<strong>它从实内积空间 <script type="math/tex">V</script> 映射到 <script type="math/tex">V</script> 自身</strong>，且保证变换前后内积不变。</p>
</blockquote>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>关系</th>
<th>表达式</th>
<th>矩阵要求</th>
<th>记号</th>
</tr>
</thead>
<tbody>
<tr>
<td>相似</td>
<td>
<script type="math/tex">P^{-1}AP=\Lambda</script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，可逆矩阵 <script type="math/tex"> P </script>
</td>
<td>
<script type="math/tex"> A \sim \Lambda </script>
</td>
</tr>
<tr>
<td>相似</td>
<td>
<script type="math/tex">Q^{-1}AQ=\Lambda</script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，<strong>正交</strong>矩阵 <script type="math/tex"> Q </script>
</td>
<td>
<script type="math/tex"> A \sim \Lambda </script>
</td>
</tr>
<tr>
<td>合同相似</td>
<td>
<script type="math/tex">Q^{T}AQ=\Lambda</script>
</td>
<td>任意矩阵 <script type="math/tex">A</script>，<strong>正交</strong>矩阵 <script type="math/tex"> Q </script>
</td>
<td>
<script type="math/tex"> A \sim \Lambda </script>
</td>
</tr>
<tr>
<td>合同</td>
<td>
<script type="math/tex">C^{T}AC=\Lambda</script>
</td>
<td>任意实对称矩阵 <script type="math/tex">A</script>，可逆矩阵 <script type="math/tex"> C </script>
</td>
<td>
<script type="math/tex"> A ≃ \Lambda </script>
</td>
</tr>
</tbody>
</table></div>
<h3 id="3_8">3. 正负惯性指数</h3>
<ul>
<li><strong>二次型转化成标准型或规范性，正负惯性指数不变（正负平方项的个数，正负特征值的个数）</strong></li>
</ul>
<blockquote class="content-quote">
<p>配方法可以求正负惯性指数，但是不可以求特征值</p>
</blockquote>
<h3 id="4_6">4. 几何应用</h3>
<h5 id="lambda_1lambda_2lambda_3-neq-0">一、 <script type="math/tex"> \lambda_1、\lambda_2、\lambda_3 \neq 0 </script>
</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
(3正)\ \ \ \ \ \ \ \cfrac{x^2}{a^2}(+)\cfrac{y^2}{b^2}(+)\cfrac{z^2}{c^2} &= 1\  (椭球面)   \\
(2正1负)\ \ \ \ \ \ \ \cfrac{x^2}{a^2}(+)\cfrac{y^2}{b^2}(-)\cfrac{z^2}{c^2} &= 1\ (单叶双曲面) \\
(1正2负)\ \ \ \ \ \ \ \cfrac{x^2}{a^2}(-)\cfrac{y^2}{b^2}(-)\cfrac{z^2}{c^2} &= 1\ (双叶双曲面) \\[1em]
\cfrac{x^2}{a^2}(+)\cfrac{y^2}{b^2}(-)\cfrac{z^2}{c^2} &= 0\  (二次锥面) \\
\end{split}\end{equation}
</script>
</p>
<h5 id="lambda_1-neq-0lambda_2-neq-0-lambda_3-0">二、 <script type="math/tex"> \lambda_1 \neq 0、\lambda_2 \neq 0,\ \lambda_3 = 0 </script>
</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&\cfrac{x^2}{2p}(+)\cfrac{y^2}{2q} = z\ (椭球抛物面，pq同号) \ \ \ \ \ \ \ &\cfrac{x^2}{a^2}(+)\cfrac{y^2}{b^2} = 1\ (椭圆柱面)\\
&\cfrac{x^2}{2p}(-)\cfrac{y^2}{2q} = z\ (双曲抛物面，pq同号) \ \ \ \ \ \ \ &\cfrac{x^2}{a^2}(-)\cfrac{y^2}{b^2} = 1\ (双曲柱面)
\end{split}\end{equation}
</script>
</p>
<h5 id="lambda_1-neq-0-lambda_2-0-lambda_3-0">三、 <script type="math/tex"> \lambda_1 \neq 0,\ \lambda_2 = 0 、\lambda_3 = 0 </script>
</h5>
<p>
<script type="math/tex; mode=display">
\lambda_1 x^2 = by + cz + d\ (柱面)
</script>
</p>
<h3 id="4_7">4. 二次型标准化方法</h3>
<h4 id="1_16">（1）正交相似变化法</h4>
<p>步骤：</p>
<ol>
<li>写出二次型对应的矩阵 <script type="math/tex"> A </script>
</li>
<li>求出 <script type="math/tex"> A </script> 的所有特征值： <script type="math/tex"> \lambda_1,\lambda_2,...,\lambda_n </script>
</li>
<li>求出对应于特征值的特征向量： <script type="math/tex"> \xi_1,\xi_2,...,\xi_n </script>
</li>
<li>将特征向量 <strong>施密特正交化</strong>，再 <strong>单位化</strong>，得： 正交矩阵 <script type="math/tex">Q=(\eta_1,\eta_2,...,\eta_n)</script>
</li>
<li>作正交变换： <script type="math/tex"> x^TAx=y^T(Q^TAQ)y </script> ， <script type="math/tex"> Q^TAQ=\Lambda </script>
</li>
<li>得到标准形： <script type="math/tex"> f=\lambda_1y_1^2+\lambda_2y_2^2+...+\lambda_ny_n^2 </script>
</li>
</ol>
<h4 id="2_15">（2）拉格朗日配方法（合同）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f&=x_1^2+2x_2^2+5x_3^2+2x_1x_2+2x_1x_3+6x_2x_3 \\
 &=(x_1+x_2+x_3)^2+x_2^2+4x_3^2+4x_2x_3 \\
 &=(u_1)^2+u_2^2+4u_3^2+4u_2u_3 \\
 &=(u_1)^2+(u_2+2u_3)^2 \\
 &=(y_1)^2+(y_2)^2 \\[2ex]
 &=y_1^2+y_2^2
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p>
<script type="math/tex; mode=display">
含有平方项，先配方 \\[1ex]
不含平方项，但含有 x_1x_2 \\[1ex]
x_1x_2=(y_1+y_2)(y_1-y_2)
=y_1^2-y_2^2
</script>
</p>
</blockquote>
<h3 id="5_3">5. 正定矩阵</h3>
<p>定义： <script type="math/tex">n</script> 元二次型 <script type="math/tex">f(\pmb{x})=\pmb{x}^T\pmb{A}\pmb{x}</script>，若对任意的 <script type="math/tex">\pmb{x}\neq0</script>，都有 <script type="math/tex">f(\pmb{x})=\pmb{x}^T\pmb{A}\pmb{x}>0(值)</script>，则称 <script type="math/tex">f(\pmb{x})</script> 为正定二次型，称二次型的对应矩阵 <script type="math/tex">A</script> 为正定矩阵。</p>
<h4 id="1_17">（1）前提</h4>
<p>
<script type="math/tex; mode=display">
A=A^T
</script>
</p>
<h4 id="2_16">（2）充要条件</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&\ n\ 元二次型\ f(\pmb{x})=\pmb{x}^T\pmb{A}\pmb{x}\ 正定 \\[1ex]
⟺&\ A\ 的特征值\ \lambda_i>0 \\[1ex]
⟺&\ f(\pmb{x})\ 的正惯性指数\ p=n \\[1ex]
⟺&\ 存在可逆矩阵\ D，使得\ A=D^TD \\[1ex]
⟺&\ A\ 与\ E\ 合同 \\[1ex]
⟺&\ A\ 的全部顺序主子式均大于\ 0
\end{split}\end{equation}
</script>
</p>
<blockquote class="content-quote">
<p>常用判别方法：<script type="math/tex">A\ 的特征值\ \lambda_i>0</script>
</p>
</blockquote>
<h4 id="3_9">（3）必要条件</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&(1)\ \ \ \ a_{ii} > 0\ \ \ \ \ (对角线上的元素) \\[1ex]
&(2)\ \ \ \ |A| > 0
\end{split}\end{equation}
</script>
</p>
<h4 id="4_8">（4）重要条件</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
A\ 为正定矩阵
&⟹kA&为正定矩阵 \\
&⟹A^T=A&为正定矩阵 \\
&⟹A^{-1}&为正定矩阵 \\
&⟹A^*&为正定矩阵 \\
&⟹A^m&为正定矩阵 \\
&⟹C^TAC&\ \ \ (C可逆)为正定矩阵 \\[2em]
A,B\ 为正定矩阵
&⟹A+B&为正定矩阵 \\
&⟹
\left[\begin{array}{c c}
A & O \\
O & B
\end{array}\right]
&为正定矩阵 \\
且\ AB=BA
&⟹AB&为正定矩阵 \\
且\ A\ 为正交矩阵
&⟹A=E \\
\end{split}\end{equation}
</script>
</p>
<h3 id="6_2">6. 负定矩阵</h3>
<p>偶数阶顺序主子式全部大于0，奇数阶顺序主子在全部小于0</p>
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